Homological properties of modules over semigroup algebras

Let S be a semigroup. In this paper we investigate the injectivity of ?¹(S) as a Banach right module over ?¹(S). For weakly cancellative S this is the same as studying the flatness of the predual left module c₀(S). For such semigroups S, we also investigate the projectivity of c₀(S). We prove that for many semigroups S for which the Banach algebra ?¹(S) is non-amenable, the ?¹(S)-module ?¹(S) is not injective. The main result about the projectivity of c₀(S) states that for a weakly cancellative inverse semigroup S, c₀(S) is projective if and only if S is finite.     

Behaving sequences

Let S be a sequence over an additively written abelian group. We denote by h(S) the maximum of the multiplicities of S, and by ∑(S) the set of all subsums of S. In this paper, we prove that if S has no zero-sum subsequence of length in [1,h(S)], then either |∑(S)|?2|S|−1, or S has a very special structure which implies in particular that ∑(S) is an interval. As easy consequences of this result, we deduce several well-known results on zero-sum sequences.     

The automorphism group of a class of semi-groups

Let (S,·) be a semi-group having the following properties: (1)S=∪S _α where α is in some index setI andS _α are subgroups isomorphic to each other, (2)S _α∩S _β=Ø, a void set for α≠β and (3) the identity ofS _α is a left identity ofS for each α inI. Then the automorphism group Aut (S) ofS is studied from the point of category theory. It is proved that Aut (S) is determined by Aut (S _α) and right multiplications by the identities of groupsS _α.     

Edge-colorings of cubic graphs with elements of point-transitive Steiner triple systems

A cubic graph G is S-edge-colorable for a Steiner triple system S if its edges can be colored with points of S in such a way that the points assigned to three edges sharing a vertex form a triple in S. We show that a cubic graph is S-edge-colorable for every non-trivial affine Steiner triple system S unless it contains a well-defined obstacle called a bipartite end. In addition, we show that all cubic graphs are S-edge-colorable for every non-projective non-affine point-transitive Steiner triple system S.     

Conjecture of Li and Praeger concerning the isomorphisms of Cayley graphs of A5

LetG be a finite group, andS a subset ofG \ |1| withS =S ^?1. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS ^α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S ^?1 and | S | ≤m is Cl. In this paper we prove that the alternating groupA ₅ is a 4-Cl-group, which was a conjecture posed by Li and Praeger.     

Generators and relations for subsemigroups via boundaries in Cayley graphs

Given a finitely generated semigroup S and subsemigroup T of S, we define the notion of the boundary of T in S which, intuitively, describes the position of T inside the left and right Cayley graphs of S. We prove that if S is finitely generated and T has a finite boundary in S then T is finitely generated. We also prove that if S is finitely presented and T has a finite boundary in S then T is finitely presented. Several corollaries and examples are given.     

On the number of zero-sum subsequences

For a sequence S of elements from an additive abelian group G, let f(S) denote the number of subsequences of S the sum of whose terms is zero. In this paper we characterize all sequences S in G with f(S)>2^|S|-2, where |S| denotes the number of terms of S.     

Decomposability for stable processes

We characterize all possible independent symmetric α-stable (SαS) components of an SαS process, 0<α<2. In particular, we focus on stationary SαS processes and their independent stationary SαS components. We also develop a parallel characterization theory for max-stable processes.     

Perfect Bricks of Every Size

We answer an open question from a previous investigation related to numerical semigroups. For integers k,n≥2 we prove the existence of a numerical semigroup S and a relative ideal I such that the size of the minimal generating set for I is k, the size of the minimal generating set for the dual, S?I, is n, and the size of the minimal generating for the ideal sum I+(S?I) is nk. Further, we outline a method for proving that S is symmetric and S+(S?I)=S?{0}. The primary tool in this investigation is the Apery set of S relative to the multiplicity of S.     

On algebraic semigroups and monoids,II

Consider an algebraic semigroup S and its closed subscheme of idempotents, E(S). When S is commutative, we show that E(S) is finite and reduced; if in addition S is irreducible, then E(S) is contained in a smallest closed irreducible subsemigroup of S, and this subsemigroup is an affine toric variety. It follows that E(S) (viewed as a partially ordered set) is the set of faces of a rational polyhedral convex cone. On the other hand, when S is an irreducible algebraic monoid, we show that E(S) is smooth, and its connected components are conjugacy classes of the unit group.     

Periods of principal homogeneous spaces of algebraic tori

A generic torsor of an algebraic torus S over a field F is the generic fiber of a S-torsor P → T, where P is a quasi-trivial torus containing S as a subgroup and T = P/S. The period of a generic S-torsor over a field extension K/F, i.e., the order of the class of the torsor in the group H ^p1(K, S) does not depend on the choice of a generic torsor. In the paper we compute the period of a generic torsor of S in terms of the character lattice of the torus S.     

Numerical range of a matrix: some effective criteria

Algorithms are presented which decide, for a given complex number w and a given complex n×n matrix S, whether w is in the numerical range W(S) of S, whether w is a boundary point of W(S), whether w is an extreme point of W(S), whether w is a bare point of W(S), and whether w is a vertex of W(S). Further algorithms decide whether W(S) intersects a given line (or a given ray), whether W(S) is included in a given open half plane (or a given closed half plane), and, for a given real number r, whether the numerical radius ρ_s of S is > r, whether ρ_s=r, and whether ρ_s≥r. A simple effective criterion for H-stability is also given: a nonsingular H-semistable matrix S is H-stable iff the nullity of (S+S¹)S^-1(S+S¹) is twice the nullity of S+S¹. The computations involved in all these algorithms are elementary (rational operations, the max operation on pairs of real numbers, the degree of a nonzero polynomial, and the number of sign variations in the coefficients of a nonzero real polynomial), must be carried out exactly, and give exact (i.e., 100% reliable) results. Examples are worked out to illustrate the application of some of the algorithms.     

Une relation de séparation entre cocircuits d'un matroide

Let S₁′S₂′S₃′ be 3 distinct cocircuits of a matroid M on a set E. We say that S₁′ does not separate S₂′ and S₃′ when S₂′\S₁′ and S₃′\S₁′ are included in one single and the same component of the submatroid M × (E\S₁′). Our main result is: A matroid is graphic if and only if from any 3 cocircuits having a non-empty intersection there is at least one which separates the two others.     

Security in graphs

Let G=(V,E) be a graph. A set S⊆V is a defensive alliance if |N[x]∩S|?|N[x]-S| for every x∈S. Thus, each vertex of a defensive alliance can, with the aid of its neighbors in S, be defended from attack by its neighbors outside of S. An entire set S is secure if any subset X⊆S can be defended from an attack from outside of S, under an appropriate definition of what such a defense implies. Necessary and sufficient conditions for a set to be secure are determined.     

Powerful alliances in graphs

For a graph G=(V,E), a non-empty set S⊆V is a defensive alliance if for every vertex v in S, v has at most one more neighbor in V−S than it has in S, and S is an offensive alliance if for every v∈V−S that has a neighbor in S, v has more neighbors in S than in V−S. A powerful alliance is both defensive and offensive. We initiate the study of powerful alliances in graphs.     

Self-maps of the product of two spheres fixing the diagonal

We compute the monoid of essential self-maps of Sn×Sn fixing the diagonal. More generally, we consider products S×S, where S is a suspension. Essential self-maps of S×S demonstrate the interplay between the pinching action for a mapping cone and the fundamental action on homotopy classes under a space. We compute examples with non-trivial fundamental actions.     

Orthonormal bases and quasi-splitting subspaces in pre-Hilbert spaces

Let S be a pre-Hilbert space. We study quasi-splitting subspaces of S and compare the class of such subspaces, denoted by Eq(S), with that of splitting subspaces E(S). In [D. Buhagiar, E. Chetcuti, Quasi splitting subspaces in a pre-Hilbert space, Math. Nachr. 280 (5-6) (2007) 479-484] it is proved that if S has a non-zero finite codimension in its completion, then Eq(S)≠E(S). In the present paper it is shown that if S has a total orthonormal system, then Eq(S)=E(S) implies completeness of S. In view of this result, it is natural to study the problem of the existence of a total orthonormal system in a pre-Hilbert space. In particular, it is proved that if every algebraic complement of S in its completion is separable, then S has a total orthonormal system.     

Bounds on a graph's security number

Let G=(V,E) be a graph. A set S⊆V is a defensive alliance if |N[x]∩S|?|N[x]-S| for every x∈S. Thus, each vertex of a defensive alliance can, with the aid of its neighbors in S, be defended from attack by its neighbors outside of S. An entire set S is secure if any subset X⊆S, not just singletons, can be defended from an attack from outside of S, under an appropriate definition of what such a defense implies. The security number s(G) of G is the cardinality of a smallest secure set. Bounds on s(G) are presented.     

Divisibility properties of power LCM matrices by power GCD matrices on gcd-closed sets

Let e and n be positive integers and S={x₁,…,x_n} a set of n distinct positive integers. For x∈S, define . The n×n matrix whose (i,j)-entry is the eth power (x_i,x_j)^e of the greatest common divisor of x_i and x_j is called the eth power GCD matrix on S, denoted by (S^e). Similarly we can define the eth power LCM matrix [S^e]. Bourque and Ligh showed that (S)∣[S] holds in the ring of n×n matrices over the integers if S is factor closed. Hong showed that for any gcd-closed set S with |S|≤3, (S)∣[S]. Meanwhile Hong proved that there is a gcd-closed set S with max_x∈S{|G_S(x)|}=2 such that (S)?[S]. In this paper, we introduce a new method to study systematically the divisibility for the case max_x∈S{|G_S(x)|}≤2. We give a new proof of Hong’s conjecture and obtain necessary and sufficient conditions on the gcd-closed set S with max_x∈S{|G_S(x)|}=2 such that (S^e)|[S^e]. This partially solves an open question raised by Hong. Furthermore, we show that such factorization holds if S is a gcd-closed set such that each element is a prime power or the product of two distinct primes, and in particular if S is a gcd-closed set with every element less than 12.     

Generalized characters of the symmetric group

Normalized irreducible characters of the symmetric group S(n) can be understood as zonal spherical functions of the Gelfand pair (S(n)×S(n),diagS(n)). They form an orthogonal basis in the space of the functions on the group S(n) invariant with respect to conjugations by S(n). In this paper we consider a different Gelfand pair connected with the symmetric group, that is an “unbalanced” Gelfand pair (S(n)×S(n−1),diagS(n−1)). Zonal spherical functions of this Gelfand pair form an orthogonal basis in a larger space of functions on S(n), namely in the space of functions invariant with respect to conjugations by S(n−1). We refer to these zonal spherical functions as normalized generalized characters of S(n). The main discovery of the present paper is that these generalized characters can be computed on the same level as the irreducible characters of the symmetric group. The paper gives a Murnaghan-Nakayama type rule, a Frobenius type formula, and an analogue of the determinantal formula for the generalized characters of S(n).